Numerical Analysis and Computer Programming Transportation and assignment problems.įamily of surfaces in three dimensions and formulation of partial differential equations Solution of quasilinear partial differential equations of the first order, Cauchy’s method of characteristics Linear partial differential equations of the second order with constant coefficients, canonical form Equation of a vibrating string, heat equation, Laplace equation and their solutions. Linear programming problems, basic solution, basic feasible solution and optimal solution Graphical method and simplex method of solutions Duality. Uniform convergence, continuity, differentiability and integrability for sequences and series of functions Partial derivatives of functions of several (two or three) variables, maxima and minima.Īnalytic function, Cauchy-Riemann equations, Cauchy's theorem, Cauchy's integral formula, power series, representation of an analytic function, Taylor’s series Singularities Laurent’s series Cauchy’s residue theorem Contour integration. Riemann integral, improper integrals Fundamental theorems of integral calculus. Continuity and uniform continuity of functions, properties of continuous functions on compact sets. Real number system as an ordered field with least upper bound property Sequences, limit of a sequence, Cauchy sequence, completeness of real line Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Rings, subrings and ideals, homomorphisms of rings Integral domains, principal ideal domains, Euclidean domains and unique factorization domains Fields, quotient fields. Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem. Gauss and Stokes’ theorems, Green's identities. Application to geometry: Curves in space, curvature and torsion Serret-Furenet's formulae. Scalar and vector fields, differentiation of vector field of a scalar variable Gradient, divergence and curl in cartesian and cylindrical coordinates Higher order derivatives Vector identities and vector equation. Equilibrium of a system of particles Work and potential energy, friction, Common catenary Principle of virtual work Stability of equilibrium, equilibrium of forces in three dimensions. Rectilinear motion, simple harmonic motion, motion in a plane, projectiles Constrained motion Work and energy, conservation of energy Kepler’s laws, orbits under central forces. Application to initial value problems for 2nd order linear equations with constant coefficients. Laplace and Inverse Laplace transforms and their properties, Laplace transforms of elementary functions. Section order linear equations with variable coefficients, Euler-Cauchy equation Determination of complete solution when one solution is known using method of variation of parameters. Second and higher order liner equations with constant coefficients, complementary function, particular integral and general solution. Riemann’s definition of definite integrals Indefinite integrals Infinite and improper integral Double and triple integrals (evaluation techniques only) Areas, surface and volumes.Ĭartesian and polar coordinates in three dimensions, second degree equations in three variables, reduction to Canonical forms straight lines, shortest distance between two skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.įormulation of differential equations Equations of first order and first degree, integrating factor Orthogonal trajectory Equations of first order but not of first degree, Clairaut’s equation, singular solution. Real numbers, functions of a real variable, limits, continuity, differentiability, mean-value theorem, Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes Curve tracing Functions of two or three variables Limits, continuity, partial derivatives, maxima and minima, Lagrange’s method of multipliers, Jacobian. Algebra of Matrices Row and column reduction, Echelon form, congruence’s and similarity Rank of a matrix Inverse of a matrix Solution of system of linear equations Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues. Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, Linear transformations, rank and nullity, matrix of a linear transformation.
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